Conditional probability
1. Conditional probability
So far we have looked at independent probabilities, but what if the outcome of one event impacts another?2. Multiple meetings
Let's revisit our sales meeting scenario. Brian has been selected to attend a client meeting, so his name is no longer in the box. However, we now have another client who wants to meet at the same time, so we need to pick another salesperson. Brian is unavailable, so we'll pick between the remaining three. This is called sampling without replacement since we aren't replacing the name we already pulled out.3. Multiple meetings
This time, Claire is picked. The probability of this is one out of three or about 33%.4. Dependent events
This is an example of dependent events, where the outcome of the first event changes the probability of the second. The probability that Claire is picked second depends on who gets picked first.5. Dependent events
If Claire is picked first, there's 0% probability that Claire will be picked second.6. Dependent events
If someone else is picked first, there's a 33% probability that Claire will be picked second. In general, when sampling without replacement, each pick is dependent.7. Conditional probability
Conditional probability is a method used to calculate the chances of dependent events due to the probability of one event impacting another. This occurs frequently in real-life. For example, the probability of a train arriving on time, given the one before was delayed, or the probability of a movie being successful, given the director previously won an Oscar. Notice that these scenarios require some prior knowledge - if we don't know that the previous train was late, then we may not accurately estimate the probability of the current train being on time.8. Venn diagrams
We can visualize conditional probability using a Venn diagram. This is a technique used to display the possible outcomes of multiple events, and the overlap where both events can occur. Where the two events are dependent, a Venn diagram's overlap will change based on the results of the first event.9. Kitchen sales over $150
As an example, we can apply conditional probability to our online sales data to calculate the chance of an order being worth more than $150, given it is a kitchen product. There are 601 orders worth over $150 - 20 that are also kitchen products and 581 that are not - and 181 orders for kitchen products, including the 20 kitchen products over $150.10. Kitchen sales over $150
To calculate the conditional probability of orders being over $150 given that they are kitchen products, we divide the 20 orders meeting both conditions by the total number of orders, 1767, then divide this result by the proportion of all orders that are for kitchen products, which is 181 divided by 1767. As both of these values involve dividing by total orders we can cancel out the 1767, so the probability is 20 divided by 181, or 11%.11. The order of events matters
To highlight how the probability is conditional for dependent events, we can switch the order of events. Let's calculate the conditional probability of an order being for kitchen products, given the order is worth more than $150. We again divide the 20 orders meeting both conditions by the total number of orders, and this time we divide the result by the proportion of all orders that are over $150. Again we can cancel out the 1767, so the probability is 20 divided by 601, or 3.3%.12. Conditional probability formula
The general formula for conditional probability is shown here. The probability of event A, given event B, is equal to the probability of both events occurring, divided by the probability of event B.13. Let's practice!
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